Applications of $\alpha$-strongly regular distributions to Bayesian auctions

Two classes of distributions that are widely used in the analysis of Bayesian auctions are the Monotone Hazard Rate (MHR) and Regular distributions. They can both be characterized in terms of the rate of change of the associated virtual value functions: for MHR distributions the condition is that for values $v < v’$,$\phi(v’) - \phi(v) \ge v’ - v$,andforregulardistributions,$\phi(v’)

• \phi(v) \ge 0$\mathrm{.}ColeandRoughgardenintroducedtheinterpolatingclassof$\alpha$-StronglyRegulardistributions($\alpha$-SR distributions for short), for which $\varphi (v^{\mathrm{\prime }})-\varphi (v)\ge \alpha (v^{\mathrm{\prime }}-v)$, for $0\le \alpha \le 1$. In this paper, we investigate five distinct auction settings for which good expected revenue bounds are known when the bidders’ valuations are given by MHR distributions. In every case, we show that these bounds degrade gracefully when extended to $\alpha$-SR distributions. For four of these settings, the auction mechanism requires knowledge of these distribution(s) (in the other setting, the distributions are needed only to ensure good bounds on the expected revenue). In these cases we also investigate what happens when the distributions are known only approximately via samples, specifically how to modify the mechanisms so that they remain effective and how the expected revenue depends on the number of samples.